#109572

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

904 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,4,3)(13,14)(15,21)(16,19)(17,20)(18,23)(22,25)(24,26)(27,28);(5,6,7,9,8,10,11,12)(13,15)(14,18,20,21)(16,22)(17,23)(19,25,26,27)(24,28);(1,3,4,2)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,g^2=i,b^a=bd,c^a=ce,g^a=gh,d^b=dh,e^b=eh,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16768);